Is there a way to loop in some way, the w slider on 4D nodes?
You can use a 3d texture and animate the movement of the coordinate space.
I have a sample scene here with a large plane that is just meant to show how the coordinate plane is moving, and a cube representing your object.  Right now they both have Object Coordinates showing.

Using a Mapping node, we can rotate the coordinate space.  If you use these coordinates to generate noise and animate the z rotation, the noise will be rotating about your object.

If you don't want rotation but instead a more linear movement, you can add a second mapping node in front to offset the origin and thus offset the point of rotation.  Doing them both in one mapping node still rotates from the center.

From here, every 360 degree rotation still puts you right back where you started, but indeed you could map out any movement you wanted.
If you find the effect is not to your liking from some angles, I bet it would be possible to combine this with a W value that climbs to a certain value and falls back down, and it may not appear such a sudden change.